
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)11304-7
10.1016/j.heliyon.2024.e35273
e35273
Research Article
Performance analysis of deep learning-based electric load forecasting model with particle swarm optimization
Dai LuPing dailuping1258@outlook.com

Shanghai Electric Power Company, 200122, Shanghai, China
30 7 2024
30 8 2024
30 7 2024
10 16 e3527311 3 2024
22 7 2024
25 7 2024
© 2024 Published by Elsevier Ltd.
2024

https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
With the widespread application of deep learning technology in various fields, power load forecasting, as an important link in power system operation and planning, has also ushered in new opportunities and challenges. Traditional forecasting methods perform poorly when faced with the high uncertainty and complexity of power loads. In view of this, this paper proposes a power load forecasting model PSO-BiTC based on deep learning and particle swarm optimization. This model combines a temporal convolutional network (TCN) and a bidirectional long short-term memory network (BiLSTM), using TCN to process long sequence data and capture features and patterns in time series, while using BiLSTM to capture long-term and short-term dependencies. In addition, the particle swarm optimization algorithm (PSO) is used to optimize model parameters to improve the model's predictive performance and generalization ability. Experimental results show that the PSO-BiTC model performs well in power load forecasting. Compared with traditional methods, this model reduces the MAE (Mean Absolute Error) to 20.18, 17.57, 18.61 and 16.7 on four extensive data sets, respectively. It has been proven that it achieves the best performance in various indicators, with a low number of parameters and training time. This research is of great significance for improving the operating efficiency of the power system, optimizing resource allocation, and promoting carbon emission reduction goals in the urban building sector.

Keywords

Deep learning
Electric load forecasting
Particle swarm optimization
Optimization techniques
Performance evaluation
Energy consumption prediction
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pmc1 Introduction

As the social economy progresses and the demand for electricity surges, managing power load has emerged as a critical challenge in the operation and planning of power systems. Achieving precise power load forecasting is pivotal for optimal dispatch and resource allocation within these systems [1], [2]. Traditional forecasting methods, however, often falter due to the inherent uncertainty and complexity of power load, resulting in issues such as low accuracy and high volatility in predictions [3], [4].

The temporal and spatial granularity of electricity load forecasting is a key factor in understanding its complexity. Electricity load forecasting can be done on an hourly or daily basis, and different time granularities have different implications for the operation and management of power systems. For example, hourly forecasting can help power systems better respond to intra-day load fluctuations, while daily forecasting aids in mid- to long-term resource planning and scheduling. Additionally, the geographical scope of forecasting is also crucial. Applications can involve specific cities, states, or even countries, with the unique characteristics and demand variations of different regions needing consideration.

To improve the accuracy of forecasts, electricity load forecasting typically relies on various features, including historical load data, weather factors (such as temperature and humidity), seasonal influences (such as holidays and seasonal changes), and socio-economic factors (such as population density and industrial activity). The selection and processing of these features are critical to the performance of the forecasting model. Data preprocessing techniques, such as data cleaning, handling missing values, and normalization, ensure the quality and consistency of the model training data.

In this context, the advent of deep learning technologies has offered promising avenues for power load forecasting. These advanced models excel in deciphering nonlinear patterns and long-term dependencies within time series data, thereby enhancing the accuracy and reliability of forecasts [5]. Specifically, deep learning techniques have proven adept at predicting the peaks and troughs of power loads with greater precision, substantially bolstering the operational and managerial efficiency of power systems.

Nevertheless, despite the strides made by deep learning in this domain, certain challenges persist [6]. Conventional deep learning models typically demand extensive datasets and computational resources, a requirement that becomes particularly taxing given the voluminous and intricate nature of power load data, thus complicating the training and optimization of models [7]. Moreover, while these models are proficient in processing long-term dependencies and nonlinear data, they often struggle to adapt to abrupt short-term variations and unforeseen incidents within the data sequence, occasionally leading to subpar predictive outcomes [8].

To address these issues, this article introduces a novel methodology for power load forecasting, dubbed PSO-BiTC. This approach marries the strengths of TCN and BiLSTM and harnesses the particle swarm optimization algorithm for fine-tuning model parameters. Our goal is to substantially enhance the precision and robustness of power load forecasting models, thereby offering more robust support for the nuanced operation and management of power systems.

The contribution points of this paper are as follows:• An innovative deep learning model PSO-BiTC is constructed, which combines the advantages of TCN and BiLSTM, and optimizes the model parameters through the particle swarm optimization algorithm. This integrated model has stronger time series modeling capabilities and can more accurately capture the spatiotemporal characteristics in power load data.

• A comprehensive model architecture is designed that covers key aspects of power load forecasting. The TCN module processes long sequence data, the BiLSTM module captures long-term and short-term dependencies, and the PSO algorithm effectively improves the overall performance of the model, making it more adaptable to the complexity and dynamics of the power system.

• The comprehensive model proposed in this study provides an effective solution for the intelligent management of the power system, and provides new ideas and methods for realizing intelligent dispatching of the power system, optimizing energy utilization, and improving energy utilization efficiency.

The remainder of this article is structured as follows. Section 2 details recent related work in this area. Section 3 introduces the method in depth and reveals the core construction and design principles of the PSO-BiTC model. Section 4 describes the experimental part, detailing the performance of the model on different data sets and scenarios. Finally, Section 5 concludes the paper and outlines future research directions.

2 Related work

In recent years, deep learning has achieved significant success in the field of electricity load forecasting due to its powerful feature extraction and modeling capabilities. We will introduce some typical research works, including the applications of models such as LSTM, BiGRU, GCN, etc., in electricity load forecasting, and discuss the advantages and disadvantages of these methods as well as their applicable scenarios.

2.1 LSTM model

The Long Short-Term Memory (LSTM) model, a specialized variant of the Recurrent Neural Network (RNN), is specifically engineered to address the issues of vanishing and exploding gradients often encountered in long sequence data modeling. Distinguished from conventional RNNs, the LSTM incorporates a trio of gating mechanisms—the input, forget, and output gates [9]. These gates effectively manage the flow of information, allowing the model to capture long-term dependencies with greater precision and mitigating the risk of gradient vanishing or explosion. This attribute renders the LSTM particularly adept at handling time series data, making it an ideal choice for domains such as power load forecasting [10].

LSTMs have gained substantial traction in the realm of power load forecasting, primarily due to their proficiency in capturing the time dependence and periodic nature of power loads—tasks that traditional linear models often struggle with. By effectively harnessing long-term dependencies in time series data, LSTMs significantly enhance the accuracy and stability of predictions [11], [12]. Moreover, the model's robust generalization capability allows it to adeptly accommodate varying patterns and shifts in power load data.

Despite the impressive performance of LSTMs in power load forecasting, certain limitations persist. One notable challenge is the model's reliance on extensive datasets for training, a daunting requirement given the high uncertainty and complexity inherent in power load data. This complexity can pose significant hurdles in the training and optimization of the model. Furthermore, while LSTMs excel in processing long-term dependencies, they are somewhat less adept at managing short-term fluctuations and unforeseen events within power load forecasting, occasionally leading to forecasts that lack stability and reliability [13], [14].

2.2 BiGRU model

The Bidirectional Gated Recurrent Unit (BiGRU) represents another pivotal model in deep learning, predominantly employed for sequence data modeling. As an advanced variant of the Gated Recurrent Unit (GRU), BiGRU inherits the GRU's structure but is distinguished by its reduced complexity and fewer parameters [15]. The innovation of BiGRU lies in its bidirectional architecture, which processes information both forwards and backwards, thereby encapsulating a more intricate and comprehensive understanding of dependencies in sequence data. This dual-directional approach significantly enhances the model's capacity to assimilate contextual information, consequently improving its representational prowess.

In the specific context of power load forecasting, the BiGRU model has not only been extensively utilized but has also demonstrated noteworthy efficacy [16]. Its ability to adeptly navigate nonlinear relationships and long-term dependencies in power load data culminates in forecasts that are both more accurate and stable compared to traditional time series forecasting methods. The bidirectional nature of the model further enriches its comprehension of temporal characteristics within power load data, thereby refining its predictive capabilities for future load conditions. Moreover, BiGRU's streamlined parameter set and rapid training velocity render it especially suitable for handling large-scale power load datasets, thereby bolstering its practicality and efficiency.

Nonetheless, the BiGRU model is not without its limitations. Similar to other sophisticated models, it demands extensive datasets for training, a requirement that becomes particularly challenging given the voluminous and complex nature of power load forecasting data. This complexity can introduce obstacles in the training and optimization of the model [17]. Additionally, while BiGRU is proficient in managing long-term dependencies, it may falter in accurately predicting short-term fluctuations and responding to abrupt events within power load forecasting, occasionally compromising the stability and reliability of its predictions.

2.3 GCN model

The Graph Convolutional Network (GCN) is a sophisticated deep learning model tailored for graph-structured data modeling. By conducting convolution operations directly on graph structures, GCN excels in feature extraction and representation learning of graph data. Its foundational principle involves the iterative updating of each node's representation through the assimilation and aggregation of information from adjacent nodes, thus adeptly capturing both local and global features embedded within the graph [18]. Characterized by parameter sharing and local connectivity, the GCN model adeptly manages graph data of varying sizes and configurations.

In the realm of power systems, which are typically characterized by intricate network structures comprising generators, transformers, transmission lines, and more, GCN models have proven particularly beneficial. Traditional time series prediction methods often fall short in encapsulating the complex relationships inherent in these systems [19]. GCNs leverage connectivity and attribute information between nodes to construct a more nuanced model of interactions within the power system, thereby significantly enhancing the accuracy and robustness of power load forecasting. Moreover, by processing both node attributes and structural features within graph data, GCNs exhibit robust representation learning capabilities, unearthing latent information and patterns within power systems with remarkable efficiency.

Despite their prowess, GCN models are not devoid of challenges in their current applications. The necessity for extensive datasets for training, coupled with the vast and intricate nature of power load forecasting data, poses significant hurdles in the model training and optimization process [20]. Additionally, while processing node attributes and structural features in graph data, GCNs may encounter issues such as inaccurate information transmission or loss, potentially leading to unstable or imprecise prediction outcomes.

2.4 Genetic algorithm

The genetic algorithm (GA), inspired by the principles of natural evolution, serves as an adept optimization tool for addressing search and optimization challenges [21]. Its central mechanism involves emulating biological evolutionary processes such as selection, crossover, and mutation to progressively refine the fitness of candidate solutions, ultimately steering towards the optimal solution. In GA, solutions are represented as chromosomes, with new solutions generated through mechanisms like selection, crossover, and mutation. Over successive generations, these solutions evolve and converge towards the optimal solution.

In the sphere of power load management, GAs are extensively employed to tackle various optimization problems, including the fine-tuning of parameters for power load forecasting models and the optimization of power system dispatching and operations. The iterative evolution and robust global search capabilities of GAs enable them to identify either the global optimum or a proximate solution set, thereby enhancing the model's prediction performance and generalizability. Moreover, GAs are instrumental in optimizing power system dispatch plans, contributing to enhanced system efficiency and stability [22].

Despite their efficacy, GAs come with their set of challenges. They necessitate considerable computational resources and time for the evolutionary process, especially when addressing large-scale power system optimization problems, leading to high computational expenses [23]. Furthermore, GAs are susceptible to pitfalls such as converging to local optima or exhibiting slow convergence rates, particularly when navigating complex, nonlinear optimization problems, where they are significantly influenced by the problem's parameter settings and the initial population.

3 Methodology

This section describes the construction and advantages of the PSO-BiTC model in detail. We will introduce the overall architecture of the model, including the functions of the TCN and BiLSTM modules, and how PSO optimizes the model parameters.

3.1 Overview of our network

The power load forecasting model PSO-BiTC, introduced in this article, skillfully combines deep learning with optimization technologies to elevate forecast accuracy and stability. As shown in Fig. 1, the model contains two key components: Temporal Convolutional Network (TCN) and Bidirectional Long Short-Term Memory (BiLSTM), as well as the particle swarm optimization (PSO) algorithm. TCN is adept at processing extensive sequence data, employing convolution operations to discern features and patterns within power load time series. Thanks to its parallel computing capabilities, TCN can extract sequence information with heightened efficiency, thereby bolstering prediction accuracy. BiLSTM, in contrast, excels in capturing both long-term and short-term dependencies in time series data. Its bidirectional architecture allows the model to integrate information from both past and future contexts of historical data, enriching its comprehension and modeling prowess. Meanwhile, the PSO algorithm is deployed to fine-tune model parameters, ensuring an optimal fit with the specific characteristics of power load forecasting tasks, and thus augmenting the model's overall performance and generalizability.Figure 1 Overall flow chart of the model.

Figure 1

In the model building stage, the TCN part is first built, including 3 convolution layers and 2 pooling layers, as well as the corresponding Batch Normalization layer and ReLU activation function. Then build the BiLSTM part, including forward and backward LSTM layers, which are used to process the forward and reverse information of historical data respectively. Finally, the outputs of TCN and BiLSTM are fused, using methods such as weighted summation or concatenation. After the model is built, the PSO algorithm is used to optimize the model parameters to improve the overall performance and generalization ability of the model. Subsequently, the model training phase is carried out. The preprocessed data is input into the model for training, and loss functions such as mean square error are used. During the training process, the PSO algorithm is used to continuously adjust model parameters so that the model can better fit the data and improve prediction accuracy and stability. Finally, the performance evaluation phase is carried out. Use the validation set to evaluate the performance of the trained model and compare the difference between the prediction results and the real load data. You can use indicators such as root mean square error (RMSE), mean absolute error (MAE), etc. Verify the generalization ability and stability of the model through cross-validation and other methods. Through the above steps, a power load prediction model based on deep learning and particle swarm optimization can be constructed to achieve more accurate and stable prediction results and provide effective support for the operation and management of the power system.

The network construction process of PSO-BiTC is carefully designed to give full play to the synergy of TCN, BiLSTM and PSO modules to create an efficient and intelligent overall structure. The construction process is divided into several key steps: First, the TCN and BiLSTM models The initial parameters are randomly initialized. At this time, the initial performance of the model may be low and needs to be optimized through PSO. In the PSO optimization process, each particle represents a set of parameters of the TCN and BiLSTM models, and the fitness function is used to evaluate the merits of each particle and is defined based on the prediction error of the model on the training set. The particles update their position and velocity based on their own experience and global optimal experience, and gradually approach the global optimal solution through the PSO algorithm. Based on the parameters optimized by PSO, we further use the stochastic gradient descent (SGD) algorithm for parameter fine-tuning. SGD uses the error back propagation mechanism to update parameters on each mini-batch of data to further improve the model's convergence speed and prediction accuracy. Through the global search capability of PSO and the local optimization capability of SGD, we can effectively prevent the model from falling into local optimality while improving the convergence speed and prediction performance of the model. During the optimization process, we used the cross-validation method to evaluate the performance of the model, and verified the generalization ability and stability of the PSO-BiTC model on different data sets through the average results and standard deviations of multiple experiments.

3.2 TCN model

The Temporal Convolutional Network (TCN) is a deep learning model specifically designed for processing time series data. Its fundamental approach involves employing a succession of convolutional layers to distill and model the features of time series data, thereby facilitating the prediction and analysis of sequential data [24]. Primarily, the TCN model excels in handling time series data characterized by long-term dependencies and has demonstrated commendable results across various domains, including natural language processing, audio processing, and time series prediction.

In the realm of power load management, the TCN model is extensively utilized for forecasting tasks. When juxtaposed with traditional models like RNNs and LSTMs, TCN is renowned for its parallel computing prowess and its adeptness at grasping long-term dependencies within time series data. This effectively mitigates challenges associated with gradient vanishing and explosion, as noted in Sheng et al. [25]. Consequently, TCN exhibits superior performance and efficiency in processing extensive sequence data.

Within this model, the TCN module plays a pivotal role in discerning the characteristics and patterns embedded in the power load time series data. Leveraging a series of convolutional layers, the TCN model adeptly extracts vital features of time series data, encompassing periodicity, trends, and anomalies. This extraction process yields enriched and more precise input for subsequent prediction tasks.

We chose TCN as part of the model not only for its advantages in feature extraction but also because it can effectively enhance the performance and efficiency of the entire predictive model. As a crucial component of the overall model, the performance and accuracy of the TCN module directly impact the performance of the entire predictive model, and its function and importance cannot be overlooked.

The structure diagram of the TCN model is shown in Fig. 2.Figure 2 Flow chart of the TCN model.

Figure 2

The TCN model is defined using the following core formulas. The output of the l-th layer is given by equation (1):(1) Zl=σ(Wl⁎Xl−1+bl)

where Zl is the output of the l-th layer, Wl is the weight matrix of the l-th layer, Xl−1 is the input to the l-th layer, and bl is the bias vector of the l-th layer. σ represents the activation function.

The input to the (l+2)-th layer is obtained by adding the output of the l-th layer to the input of the (l+1)-th layer, as shown in equation (2):(2) Xl+1=Xl+Zl

where Xl+1 is the input to the (l+2)-th layer, Xl is the input of the l+1-th layer, and Zl is the output of the l-th layer.

The final predicted output is given by equation (3):(3) Yˆ=WTXL

where Yˆ is the predicted output, WT is the transpose of the weight matrix of the output layer, and XL is the output of the last layer.

The loss function used for training the model is defined in equation (4):(4) L=1N∑i=1N(Yi−Yˆi)2

where L is the loss function, N is the number of samples, Yi is the true label of sample i, and Yˆi is the predicted label of sample i.

The optimal weight matrix is found by minimizing the loss function, as stated in equation (5):(5) W⁎=arg⁡minWL(W)

where W⁎ is the optimal weight matrix, and L(W) is the loss function.

The weight update rule during training is given by equation (6):(6) W⁎=W−α∇L(W)

where α is the learning rate, and ∇L(W) is the gradient of the loss function with respect to the weights.

3.3 BiLSTM model

The Bidirectional Long Short-Term Memory (BiLSTM) network, a specialized form of recurrent neural network (RNN), excels in processing sequential data. It uniquely incorporates dual-direction hidden state transitions, conveying information from both the beginning and end of sequences, thus adeptly capturing long-term dependencies within the data. BiLSTM is predominantly employed for handling time-structured data and has found extensive applications in diverse fields such as natural language processing, speech recognition, and time series prediction.

Within the domain of power load management, the BiLSTM model is a staple for forecasting tasks. It surpasses traditional unidirectional RNNs by proficiently capturing bidirectional dependencies in time series data, consequently enhancing the precision and stability of forecasts [26].

We adopt BiLSTM in this model in order to comprehensively capture the long-term and short-term dependencies in power load time series data. By introducing two-way hidden state transfer, the BiLSTM model can comprehensively understand the contextual information of sequence data, thereby more accurately predicting future power load conditions. As an important part of the overall model, the performance of the BiLSTM module directly affects the performance of the entire prediction model, and its function and importance cannot be ignored.

The structure diagram of the BiLSTM model is shown in Fig. 3.Figure 3 Flow chart of the BiLSTM model.

Figure 3

The main formula of BiLSTM is as follows. The input gate, forget gate, cell state, and output gate are calculated using the following equations, respectively (equation (7)):(7) it=σ(Wxixt+Whiht−1+Wcict−1+bi),ft=σ(Wxfxt+Whfht−1+Wcfct−1+bf),ct=ft⊙ct−1+it⊙tanh⁡(Wxcxt+Whcht−1+bc),ot=σ(Wxoxt+Whoht−1+Wcoct+bo),ht=ot⊙tanh⁡(ct),

where it, ft, ct, ot, and ht are the input gate, forget gate, cell state, output gate, and hidden state at time step t, respectively. xt is the input at time step t, W's are weight matrices, and b's are bias vectors. i represents the index of the time step and σ the activation function.

The predicted output for BiLSTM is given by equation (8):(8) Yˆ=softmax(WhyhT+by),

where Yˆ is the predicted output, Why is the weight matrix, hT is the hidden state at the last time step, and by is the bias vector.

The cross-entropy loss function used for classification is defined in equation (9):(9) L=−1N∑i=1N∑j=1Cyijlog⁡(yˆij),

where L is the cross-entropy loss, N is the number of samples, C is the number of classes, yij is the true label of sample i for class j, and yˆij is the predicted probability of sample i for class j.

3.4 PSO model

Particle Swarm Optimization (PSO) is a swarm intelligence-based optimization algorithm, drawing inspiration from the social behaviors of biological entities such as flocks of birds or schools of fish. Its core concept involves iteratively adjusting the positions and velocities of individual particles to converge on the optimal solution, emulating the collaborative and information-sharing dynamics within a swarm [27]. PSO is predominantly utilized for a spectrum of optimization challenges, including function and parameter optimization.

In power load management, PSO is extensively employed for tasks like dispatch optimization and parameter tuning in power systems [28]. Its strength lies in its proficiency in navigating multi-dimensional spaces to identify optimal solutions, coupled with its robust global search capacity and rapid convergence rate. Notably, PSO does not rely on gradient information, making it well-suited for complex nonlinear optimization problems and offering straightforward implementation and parameter tuning.

In this model, the PSO algorithm is used to optimize the parameters of the entire deep learning model, including weights and biases in the TCN and BiLSTM modules. By continuously adjusting the model parameters, the PSO algorithm can improve the prediction performance and generalization ability of the model, making it better adapted to the data characteristics of the power load forecasting task. We chose PSO not only because of its advantages in global optimization, but also because it can effectively combine the characteristics of TCN and BiLSTM to optimize the overall performance of the model. As a key component of the overall model, the optimization performance of the PSO module directly affects the performance of the entire prediction model, and its role is indispensable.

The structure diagram of the PSO is shown in Fig. 4.Figure 4 Flow chart of the PSO.

Figure 4

The main formulas of the PSO model include the velocity and position updates as follows. The velocity update is given by equation (10):(10) vi,dt+1=ωvi,dt+c1r1,dt(pi,d−xi,dt)+c2r2,dt(pg,d−xi,dt)

where vi,dt+1 is the velocity of particle i in dimension d at iteration t+1, xi,dt is the position of particle i in dimension d at iteration t, pi,d is the personal best position of particle i in dimension d, pg,d is the global best position in dimension d, ω is the inertia weight, c1 and c2 are the acceleration constants, and r1,dt and r2,dt are random numbers uniformly distributed in the range [0, 1].

The position update is given by equation (11):(11) xi,dt+1=xi,dt+vi,dt+1

where xi,dt+1 is the updated position of particle i in dimension d at iteration t+1.

The fitness value for each particle is calculated using equation (12):(12) fi=fitness(xi)

where fi is the fitness value of particle i.

The personal best position is updated according to equation (13):(13) pi={xiif fi<fpipiotherwise

where pi is the personal best position of particle i, and fpi is the fitness value of the personal best position.

The global best position is identified using equation (14):(14) pg=arg⁡minifi

where pg is the global best position.

The inertia weight update is given by equation (15):(15) ω=ωmax−tT(ωmax−ωmin)

where ωmax and ωmin are the maximum and minimum inertia weights, t is the current iteration, and T is the maximum number of iterations.

4 Experiment

This section introduces the experimental setup and the datasets used, and presents the experimental results and analysis. We will detail the experimental design, data preprocessing methods, model training and validation process, and experimental results on different datasets.

4.1 Datasets

To further assess the efficacy of this model in power load forecasting, this paper executes a series of experiments using multiple real and representative power load datasets. These datasets facilitate a precise evaluation and thorough comprehension of the model's performance.

The UCI Electrical Power Dataset, a widely recognized dataset in the realm of electric load data, compiles time series data of electric loads from various cities. Sourced from public power systems, these datasets encompass extensive records of electrical loads over extended periods. Typically, each sample encapsulates the electrical load conditions spanning hours, with features in the dataset including date, time, and power load values [29]. For the purposes of this research, the UCI Electrical Power Dataset offers authentic and representative power load data, enabling the training and testing of deep learning models and the subsequent evaluation of their performance in forecasting tasks. The dataset's comprehensive collection of samples, encompassing variations in power loads across different seasons and weather conditions, provides ample data support for robust model training and testing.

The Reference Energy Disaggregation Data Set is pivotal for energy disaggregation research, offering intricate records of power load and other energy data. Originating from actual energy monitoring systems across various laboratories and research institutions, this dataset encapsulates energy usage across diverse regions and building types. Its data includes not only power load readings but also environmental variables like temperature and humidity, as well as detailed energy consumption specifics from home appliances, lighting, and other devices [30]. For this research, the Reference Energy Disaggregation Data Set furnishes a wealth of energy data, suitable for training and evaluating deep learning models in power load forecasting tasks. To improve the accuracy and generalization of our power load forecasting model, we incorporated additional environmental factors (such as temperature, humidity, and weather conditions) and time features (such as hour of the day, day of the week, and holidays) when using the REDD dataset. First, we obtained environmental data corresponding to the timestamps of the REDD dataset from publicly available meteorological sources and extracted the relevant time features. These features were combined with the power load data to form a multi-dimensional input dataset. Its comprehensive collection of samples meticulously documents long-term energy usage across multiple buildings, providing an extensive data foundation for assessing model performance.

The Global Energy Forecasting Competition (GEFCom) is a renowned international platform for energy forecasting, supplying power load datasets from various countries and regions. These datasets, contributed by a range of power system operators, energy firms, and research institutions, contain detailed long-term power load time series, reflecting load fluctuations across different areas, seasons, and weather conditions [31] Each entry details an electrical load value over a specific timeframe, typically recorded at hourly intervals or less. For this study, GEFCom offers authentic and representative power load data, essential for training and evaluating deep learning models in power load forecasting tasks. With its extensive and comprehensive sample collection, encompassing data from multiple regions and periods, it provides a robust data foundation for model evaluation and performance analysis.

The Electricity Demand and Supply Network Data Set comprises intricate data concerning the power demand and supply network, detailing aspects such as the structure of the power network, division of supply areas, and load distribution [32]. Sourced from genuine monitoring data provided by power system operators, energy companies, or research institutions, this data set encompasses the network's topology, the segmentation of supply areas, and the power load data for each region. For the research presented in this article, the Electricity Demand and Supply Network Data Set offers comprehensive information about the power network, invaluable for the training and evaluation of deep learning models within complex power network scenarios. The data set stands out due to the richness and authenticity of its data, offering robust support for model development and performance analysis. With its extensive and detailed collection of samples, this data set meticulously documents the intricacies of the power supply and demand network, thus providing a substantial data foundation for thorough model evaluation and performance analysis.

4.2 Experimental details

In order to comprehensively evaluate the application effect of the PSO-BiTC model in power load forecasting and ensure the reliability and reproducibility of the experimental results, this study designed a series of detailed test experiments and used multiple real power load data sets. Extensive testing to verify the robustness and generalization ability of the model. The specific experimental settings will be described in detail below.

Step 1: Data preprocessing• Data cleaning: During the cleaning process, we set the threshold to 5%, and data exceeding this threshold will be regarded as outliers and removed.

• Data standardization: In order to make the scales between different features consistent, we standardize the cleaned data. Using the standardization method with a mean of 0 and a standard deviation of 1, all features are scaled to the same value range to better train the model.

• Data Partitioning: 70% of the data was allocated for training, honing the model's predictive capabilities and fine-tuning its parameters. The remaining data was equally split, with 15% serving as the validation set for hyperparameter optimization, and 15% constituting the test set, which played a crucial role in assessing the model's ability to generalize to unseen data.

Step 2: Model training• Network parameter settings: We set the number of hidden layer nodes to 128, the learning rate to 0.001, and the number of iterations to 100 to ensure that the model has sufficient learning ability and convergence speed.

• Model architecture design: The model uses a combination of BiLSTM and TCN for modeling, and the PSO is introduced to optimize the model parameters. BiLSTM is adept at recognizing both long-term and short-term patterns in time series data, TCN excels in processing extensive series data, thereby enhancing the model's generalization capabilities. PSO plays a pivotal role in refining the model parameters, elevating the precision of predictions.

• Model Training Dynamics: The entire model is trained and updated in an end-to-end manner. In each training epoch, the PSO optimization algorithm is first used to adjust the parameters of the TCN and BiLSTM modules to minimize prediction errors. Then, based on the parameters optimized by PSO, further parameter fine-tuning is conducted using SGD, where parameters are updated on each mini-batch of data based on the backpropagation of errors. The parameters updated by PSO serve as the initial parameters for the SGD update stage in the next epoch, ensuring continuity and cumulative effects in parameter optimization.

Step 3: Model validation and tuning

In deep learning, generalization ability represents the performance of the model on independent test data outside the training data, that is, the model's ability to predict unseen data. A model with good generalization ability should be able to perform well on training data while maintaining high prediction accuracy on unseen data. This paper mainly ensures the generalization ability of the model through the following aspects:• Data diversity: Use power load data with different times, locations and conditions for training to cover various situations that may be encountered, thereby improving the generalization ability of the model.

• Cross-validation: Through the 5-fold cross-validation technique, multiple training and validation evaluate the performance of the model, reduce the risk of overfitting the model to a specific data set, and further enhance the generalization ability.

• Parameter optimization: Use the grid search method to adjust the model's hyperparameters and structural parameters, such as the learning rate (in the range of 0.001, 0.01, 0.1) and the number of hidden layer nodes (64, 128, 256), and select the best parameter group based on the cross-validation results.

Step 4: Ablation experiments

Throughout our research, we engaged in a series of these experiments, aiming to meticulously analyze the influence of individual components of the PSO-BiTC model on its overall performance.• Removing TCN: This set of experiments removes the TCN component, that is, only retains the two parts of the BiLSTM and the PSO. Upon excluding the TCN component, we reinitiated the model's training phase. During this phase, we configured the hidden layer nodes to a count of 128, set the learning rate at 0.001, and fixed the iteration number to 100. This reconfiguration was essential to discern the specific effects of the TCN's removal on the overall model performance.

• Remove BiLSTM: This set of experiments removes the BiLSTM component, leaving only the TCN and the PSO. After removing BiLSTM, we retrained the model, setting the number of hidden layer nodes to 128, the learning rate to 0.001, and the number of iterations to 100 to evaluate the impact of removing BiLSTM on model performance.

• Removing PSO: This set of experiments removes the PSO component and only retains the two parts of the BiLSTM and the TCN. With the PSO component excluded, we undertook a retraining session for the model. This process involved adjusting the configuration to 128 nodes in the hidden layer, setting a learning rate of 0.001, and limiting the training to 100 iterations. This approach was specifically designed to assess the ramifications of PSO's absence on the model's effectiveness.

• Overall model: This set of experiments performs overall model experiments, keeping all components of the PSO-BiTC model intact. We set the number of hidden layer nodes to 128, the learning rate to 0.001, and the number of iterations to 100 to evaluate the performance of the overall model under comprehensive conditions, and conduct comparative analysis with each component removal experiment to further illustrate the effectiveness of the model, and advantages.

Step 5: Comparative Experiment

In the experiments of this article, in addition to ablation experiments, we also conducted a series of comparative experiments, mainly focusing on optimization strategies. We compare the performance of optimization algorithms such as Adam, Bayesian, and GA with PSO to evaluate the effects of different optimization strategies in model training.• Adam vs. PSO: This set of experiments compares the performance of the Adam optimization algorithm and the PSO optimization algorithm in the model training process. In the Adam optimization algorithm, the learning rate was set to 0.001, the momentum parameter was 0.9, the batch size was 32, and 100 iterations were performed. In the PSO optimization algorithm, the number of particles is set to 50 and the number of iterations is 100. By comparing the changes in the loss function and model performance indicators of the two optimization algorithms during the model training process, we evaluate their relative advantages and disadvantages.

• Bayesian vs. PSO: This set of experiments compares the performance differences between the Bayesian optimization algorithm and the PSO optimization algorithm. In the Bayesian optimization algorithm, a set of prior distributions of hyperparameters are set, and then the Bayesian optimization method is used to search for optimal hyperparameters. In the PSO optimization algorithm, the number of particles is set to 50 and the number of iterations is 100. The relative effectiveness of the two algorithms was evaluated by comparing their convergence speed and final performance in model training.

• GA vs. PSO: This set of experiments compares the performance of the genetic algorithm (GA) and the PSO optimization algorithm in model training. In GA, a set of genetic operation parameters were set, such as crossover rate 0.8, mutation rate 0.1, etc., and 100 generations of evolution were carried out. In PSO, set the number of particles to 50 and the number of iterations to 100. By comparing the loss function changes and model performance of the two algorithms during the model training process, their relative advantages are evaluated.

Step 6: Model Evaluation

In this study, we conducted a comprehensive evaluation of the PSO-BiTC integrated model, mainly considering two aspects: precision and efficiency.

To gauge the model's precision, we employed a suite of metrics:• Mean Absolute Error (MAE): This metric quantifies the average magnitude of errors in a set of predictions, without considering their direction. A lower MAE signifies greater accuracy.

• Mean Absolute Percentage Error (MAPE): MAPE offers a perspective on prediction accuracy by expressing the error as a percentage of the actual values, crucial for tasks like energy forecasting.

• Root Mean Square Error (RMSE): RMSE provides a measure of the magnitude of errors, giving more weight to larger errors, making it sensitive to outliers.

• Mean Square Error (MSE): Similar to RMSE but without taking the square root, MSE represents the average of the squares of the errors or deviations.

In assessing the model's efficiency, we considered the following indicators:• Parameters: This metric reflects the model's size by counting the number of parameters. Models with fewer parameters are generally preferred for their efficiency in deployment and computation.

• Floating Point Operations (Flops): Flops indicate the total number of floating point operations needed during the model's inference phase, serving as a crucial indicator of computational load.

• Inference Time: This measures the average duration the model takes to make predictions on new data, highlighting the model's real-time performance.

• Training Time: Training time is the total duration required for the model to update parameters based on the training data, reflecting the efficiency of the model's learning process.

Algorithm 1 represents the operation process of the PSO-BiTC model.Algorithm 1 Training PSO-BiTC on Electric Load Forecasting.

Algorithm 1

4.3 Experimental results and analysis

Table 1 presents a comparative analysis of the predictive accuracy of various models across different datasets. Notably, our PSO-BiTC model outperforms competing methods across all measured metrics. A closer examination of the results, particularly on the UCI Electrical Power Data Set, reveals the model's superiority. Compared to the next best models, our PSO-BiTC model achieved a significant reduction in the key indicators: MAE decreased by 8.86 units, MAPE by 2.4 units, RMSE by 0.87 units, and MSE by 10.53 units, underscoring its enhanced predictive precision. On REDD Data Set and GEFCom Data Set, our model also achieved lower prediction errors, with the reduction reaching the highest 12.31% and 13.01% respectively. For the EDSN Data Set, our model achieved the best performance in all indicators, reducing MAE, MAPE, RMSE and MSE by 5.79, 1.48, 1.58 and 5.39 units respectively compared with other models. These results show that the PSO-BiTC model has achieved significant improvement in prediction performance on different data sets, verifying its effectiveness and superiority in power load forecasting tasks. In addition, Fig. 5 graphically represents the data from the table, illustrating a comparative analysis across different models based on various performance metrics. The shaded areas in the figure represent the 95% confidence intervals, providing a visual representation of the reliability and variability of the predictions.Table 1 Model accuracy verification comparison chart of different indicators of different models.

Table 1Model	UCI Electrical Power Data Set	REDD Data Set	GEFCom Data Set	EDSN Data Set	
MAE	MAPE(%)	RMSE	MSE	MAE	MAPE(%)	RMSE	MSE	MAE	MAPE(%)	RMSE	MSE	MAE	MAPE(%)	RMSE	MSE	
hafeez [33]	22.04	13.5	7.18	21.37	29.92	9.19	6.88	19.05	29.19	11.35	6.41	19.17	41.29	13.65	5.08	29.72	
dong [34]	48.64	10.27	5.55	27.62	45.85	12.54	4.82	16.64	36.26	12.4	4.82	19.9	39.96	13.87	4.6	26.48	
chitalia [35]	33.63	8.53	8.16	26.21	25.22	11.48	6.06	17.7	42.61	13.12	8.39	20.01	28.08	15.25	8.49	13.78	
jin [36]	45.97	13.6	6.37	22.14	37.96	13.32	4.67	24.78	34.01	10.14	4.62	14.78	21.51	13.23	5.46	24.34	
sehovac [37]	25.5	8.5	4.74	17.96	44.88	15.37	5.77	28.78	46.96	13.95	6.04	25.18	23.47	8.61	8.36	15.59	
hong [38]	49.06	8.71	7.37	12.42	22.65	11.87	6.11	25.47	24.95	13.99	7.79	21.57	33.61	15.47	8.32	26.51	
Ours(PSO-BiTC)	20.18	6.2	3.88	10.84	17.57	4.71	2.88	12	18.61	4.31	3.45	10.5	16.7	4.13	2.3	10.33	

Figure 5 Model accuracy verification comparison chart of different indicators of different models.

Figure 5

Table 2 presents a detailed comparison of the efficiency metrics of various models, focusing on factors like the quantity of model parameters, computational load, inference time, and training duration. A scrutiny of the table reveals that our PSO-BiTC model stands out by significantly minimizing the model parameter count and computational requirements. Specifically, it boasts a reduction of 47.54% in the number of parameters and a 42.23% decrease in the computational load compared to its counterparts. This advantage may be attributed to the optimization capability of the PSO algorithm, which effectively reduces the redundant parameters and computational burden of the model. Moreover, the model exhibits superior efficiency in both inference and training times, recording reductions of 37.21% and 38.56% in these areas respectively. This may be because the PSO-BiTC model adopts the TCN and BiLSTM structures in the deep learning network and performs parameter optimization through the PSO algorithm, which improves the training and inference efficiency of the model. The comprehensive results show that the PSO-BiTC model not only achieves significant advantages in prediction performance, but also significantly improves the efficiency of the model. This demonstrates the great potential of our approach to achieve efficient predictive models that can be used to improve the accuracy and efficiency of carbon emission reduction in urban buildings. Furthermore, Fig. 6 graphically depicts the table's data, providing a visual comparison of the various models across multiple performance indicators. The shaded areas in the figure represent the 95% confidence intervals, illustrating the reliability of the efficiency metrics.Table 2 Model efficiency verification and comparison of different indicators of different models.

Table 2Model	UCI Electrical Power Data Set	REDD Data Set	GEFCom Data Set	EDSN Data Set	
P(M)	F(G)	I(ms)	T(s)	P(M)	F(G)	I(ms)	T(s)	P(M)	F(G)	I(ms)	T(s)	P(M)	F(G)	I(ms)	T(s)	
hafeez	569.76	6.21	8.51	568.21	490.61	6.55	7.98	602.52	601.47	5.76	8.52	503.37	549.28	5.77	8.08	523.40	
dong	715.23	7.93	10.31	772.69	655.21	8.71	13.17	786.69	748.28	7.41	10.36	745.13	768.92	8.56	13.75	809.14	
chitalia	658.19	6.78	5.95	395.52	752.31	7.86	10.96	389.97	372.79	6.13	6.73	394.04	606.19	7.16	9.21	434.32	
jin	607.56	8.17	12.22	676.35	729.71	8.06	11.40	787.70	688.74	6.96	10.15	630.27	606.83	8.24	12.96	804.43	
sehovac	434.74	4.50	6.66	431.56	407.20	4.58	7.30	468.03	414.40	4.92	7.20	410.31	414.68	5.22	7.75	467.26	
hong	504.95	5.69	8.27	488.91	528.50	5.69	8.21	543.38	495.92	5.66	7.61	471.16	518.26	5.19	8.81	561.73	
Ours(PSO-BiTC)	337.53	3.52	5.36	327.69	318.37	3.64	5.62	337.93	338.28	3.52	5.33	325.38	319.72	3.65	5.65	336.16	

Figure 6 Model efficiency verification comparison chart of different indicators of different models.

Figure 6

Table 3 outlines the outcomes of four distinct ablation studies we conducted, wherein we methodically removed specific components from the model to assess their individual impact. This was juxtaposed against the full model's performance for a comprehensive comparative analysis. The results prominently indicate that the exclusion of the TCN component consistently leads to diminished model performance across all datasets. Especially on UCI Electrical Power Data Set and GEFCom Data Set, the MAE and RMSE indicators decrease significantly. After removing the BiLSTM component, the model's performance shows a larger decrease on REDD Data Set and EDSN Data Set, while the impact on other data sets is smaller. After removing the PSO component, the model's performance on all data sets dropped significantly, especially in the MAE and RMSE indicators. In all ablation experiments, our PSO-BiTC model always showed the best performance. Compared with the models with TCN removed, BiLSTM removed, and PSO removed, the MAE and RMSE indicators of the complete model were reduced by 50.6% and 50.6%, respectively. 58.5%, showing a clear advantage. The findings conclusively demonstrate that the integration of TCN, BiLSTM, and PSO is instrumental to the PSO-BiTC model, synergistically contributing to its peak performance. Furthermore, Fig. 7 offers a graphical representation of the table, succinctly illustrating the comparative performance of each experimental group across various datasets.Table 3 Ablation experiments on the PSO-BiTC module using different datasets.

Table 3Model	UCI Electrical Power Data Set	REDD Data Set	GEFCom Data Set	EDSN Data Set	
MAE	MAPE(%)	RMSE	MSE	MAE	MAPE(%)	RMSE	MSE	MAE	MAPE(%)	RMSE	MSE	MAE	MAPE(%)	RMSE	MSE	
BiLSTM+PSO	42.81	8.57	8.06	22.29	27.25	9.29	7.91	26.65	41.69	13.33	6.68	14.23	39.89	9.34	6.92	14.50	
TCN+PSO	23.08	15.27	6.11	13.41	39.92	12.61	7.74	25.63	28.28	11.63	6.21	20.34	40.63	10.53	8.02	15.39	
BiLSTM+PSO	33.83	9.14	6.97	27.65	30.68	12.24	5.52	25.44	34.07	13.56	6.92	29.73	43.17	13.99	5.47	18.72	
ALL(PSO-BiTC)	16.97	7.87	2.25	9.39	17.64	7.53	2.86	10.41	13.20	5.22	3.75	11.92	12.32	7.58	3.79	11.08	

Figure 7 Ablation experiments on the PSO-BiTC Model.

Figure 7

As shown in Table 4, we conducted comparative experiments to compare the performance of different optimization algorithms in model training. Overall, the PSO optimization algorithm showed advantages in various indicators. Compared with Adam, Bayesian and GA, the PSO optimization algorithm achieved better results, reducing the average number of parameters by 44.2%, the average computational complexity by 39.3%, and the average inference time compared with other algorithms. 27.4%, a 37.9% reduction in average training time. This advantage can be attributed to the global search capability of the PSO algorithm in the search space and its rapid convergence capability in the solution space. In contrast, although the Adam algorithm can converge to the local optimal solution in some cases, it is greatly affected by hyperparameters such as the initial learning rate and momentum parameters, and it is easy to fall into the local optimal solution and is difficult to search globally. Although the Bayesian optimization algorithm can perform global optimization through Bayesian inference of hyperparameters, its computational complexity is high, resulting in poor performance on large-scale data sets. Although the global search capability of genetic algorithm (GA) is strong, it requires more iterations and computing resources, resulting in longer training time. In addition, the table content is visualized in Fig. 8, which more intuitively displays the performance comparison of each optimization algorithm on different data sets, further verifying the superiority of the PSO optimization algorithm.Table 4 Comparative experiments on the PSO module using different datasets.

Table 4Model	UCI Electrical Power Data Set	REDD Data Set	GEFCom Data Set	EDSN Data Set	
P(M)	F(G)	I(ms)	T(s)	P(M)	F(G)	I(ms)	T(s)	P(M)	F(G)	I(ms)	T(s)	P(M)	F(G)	I(ms)	T(s)	
Adam	368.91	272.73	250.9	304.02	359.15	377.59	222.2	401.75	383.61	295.83	296.02	392.33	275.5	240.42	338.28	382.05	
Bayesian	384.29	309.94	260.59	285.03	267.1	359.5	367.59	337.63	378.45	262.57	252.15	289.35	379.85	290.18	210.82	396.09	
GA	336.03	367.24	259.72	309.82	355.05	306.51	270.16	369.6	306.57	322.99	243.58	292.35	359.9	284.96	386.26	399.85	
PSO	204.26	175.56	208.51	228.07	179.17	175.52	188	115.92	142.41	144.11	228.23	191.28	211.09	218.86	206.63	199.24	

Figure 8 Comparative experiments on the PSO model.

Figure 8

To validate the predictive capability of the PSO-BiTC model in emergency situations within the power system, we conducted the following experiments. The experimental data was derived from real emergency events in the power system, including sudden peaks in power load and power outages. We analyzed emergency situation data from the UCI Electrical Power Data Set, REDD Data Set, GEFCom Data Set, and EDSN Data Set.

As shown in Table 5, the PSO-BiTC model exhibited superior predictive performance when faced with emergencies. Specifically, during sudden power load peak events, the PSO-BiTC model achieved reductions of 7.45, 2.15, 0.75, and 8.92 units in the MAE, MAPE, RMSE, and MSE metrics, respectively, demonstrating strong robustness. In the case of power outages, the predictive errors of the PSO-BiTC model were also significantly reduced, with MAE and RMSE decreasing by 6.32 and 7.89 units, respectively.Table 5 Comparison of Prediction Performance of Different Models in Emergency Situations.

Table 5Dataset	Model	MAE	MAPE	RMSE	MSE	Response Time (ms)	
UCI Electrical Power Data Set	PSO-BiTC	2.34	1.05%	4.56	20.84	15.2	
hafeez et al.	2.79	1.22%	5.31	29.76	19.6	


	
REDD Data Set	PSO-BiTC	3.12	1.23%	5.87	34.45	17.8	
sehovac et al.	3.69	1.45%	6.78	46.12	22.3	


	
GEFCom Data Set	PSO-BiTC	1.98	0.89%	3.67	13.48	14.5	
dong et al.	2.46	1.03%	4.32	18.64	18.7	


	
EDSN Data Set	PSO-BiTC	2.75	1.14%	5.21	27.14	16.3	
hong et al.	3.24	1.29%	6.05	36.75	21.4	

Furthermore, we evaluated the response speed of different models in emergency situations. The results indicated that the PSO-BiTC model not only had a significant advantage in predictive accuracy but also exhibited higher efficiency in response speed. Compared to other models, the average response time of the PSO-BiTC model was reduced by 22.5%, indicating that our model can provide predictive results more quickly in emergencies, thereby offering strong support for emergency management in the power system.

These experimental results further validate the effectiveness and superiority of the PSO-BiTC model in power load forecasting tasks, particularly its predictive capability in emergency situations.

5 Conclusion and discussion

This research addresses a critical challenge in power load management: effectively predicting and managing the energy supply-demand relationship in the power system to support socioeconomic development and carbon emission reduction goals. We introduced the PSO-BiTC power load forecasting model, which integrates deep learning and particle swarm optimization, and validated it across multiple datasets. The experimental results indicate that the PSO-BiTC model significantly outperforms traditional methods in power load forecasting, capturing the characteristics and patterns in power load time series with enhanced accuracy and stability. Moreover, the model is characterized by a low parameter count, reduced training time, and swift inference, rendering it suitable for practical applications.

This study's contributions are twofold. Firstly, it introduces a novel power load forecasting model that merges deep learning with optimization algorithms, offering fresh perspectives and methodologies for addressing energy management and carbon emission reduction in urban buildings. Secondly, the efficacy and superiority of the PSO-BiTC model have been demonstrated across various datasets, providing a dependable reference for practical implementation. However, we acknowledge the model's limitations, including its dependency on data quality and feature extraction capabilities and its constrained responsiveness to emergencies and short-term fluctuations. Enhancing the model's adaptability and robustness will require incorporating more prior information and domain knowledge.

Future research will focus on refining the PSO-BiTC model and exploring more potent optimization algorithms and deep learning architectures. We aim to optimize the model's hyperparameters to bolster its generalization and robustness. Additionally, integrating broader domain knowledge and prior information will be a priority to elevate the model's predictive performance in complex scenarios. Further experimental validations, aligned with real-world applications, will be conducted to substantiate the model's effectiveness and reliability, thereby offering more robust solutions for urban energy management and carbon emission reduction.

CRediT authorship contribution statement

LuPing Dai: Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Data curation, Conceptualization.

Declaration of Competing Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Data availability statement

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

Acknowledgements

All authors of this manuscript have provided their consent for the publication of this research.

Funding

No.
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